Piezoelectric transducer



PIEZOELECTRIC TRANSDUCER 7 Sheets-Sheet 1 v Original Filed Oct. 13, 1954 3158623 OF5 n@ 17g/loo M 15a Feb. 2, 1965 L. A. PETERMANN 3,158,623

PIEZOELECTRIC TRANSDUCER Original Filed Oct. 13, 1954 7 Sheets-Sheet 2 VL-1 :VEJA F2-3 :155457 Byw M Feb. 2, 1%65 L. A. FETERMANN PIEZOELECTRIC TRANSDUCER Qriginal Filed Oct. 13, 1954 OUTPUT iA/PUT 7 Sheets-Sheet 3 of TTEE- al1-saveur! Fees.-

IN VEN TUR.

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PIEZOELECTRIC TRANSDUCER Original Filed Oct. 13, 1954 7 Sheets-Sheet 4 NVENTOR. Luc/EN 1Q. sTERMH/vn/ Feb. 2, i965 L. A. PETERMANN 3,168,623

PIEZOELECTRIC TRANSDUCER Original Filed 0G13. 13. 1954 7 Sheets-Sheet 5 JNVENTOR. L uc /E/v E; TER/vm NN Feb. 2, i965 L.. A. PETERMANN PIEZOELECTRIC TRANSDUCER 7 Sheets-Sheet 5 Original Filed Oct. 15, 1954 INVLVTOR. PE

ge/EN A. TERM/41W /SL M /QTTOEA/Ef Feb 2, 1955 L. A. PETERMANN PIEZOELEGTRIC TRANSDUCER 7 Sheets-Sheet 7 Original Filed Oct. 13, 1954 Rm wm f n@ m r H N E m U LY B 3,168,623 PIEZOELECTRIC TRANSDUCER Lucien A. Petermann, Geneva, Switzerland, assignor to Gulton industries, Inc., Metuchen, NJ., a corporation of New Jersey Original application Oct. 13, 1954, Ser. No. 461,979, now Patent No. 2,928,069, dated Mar. 8, 1969. Divided and this application Nov. 18, 1959, Ser. No. 853,931

31 Claims. (Cl. 179-100.611)

My invention relates to electro-mechanical transducers and their use. This is la division lof my application Serial No. 461,979, filed October 13, 1954, now Patent No. 2,928,069.

An important object of my invention is to provide a transducer with a lower resonant frequency than other transducers of comparable dimensions.

A further lobject is to provide a more sensitive transducer than other transducers of comparable dimensions.

A still fuuther object of my invention is to provide a transducer which may be produced more economically than other transducers.

A still further object lof my invention is to provide a transducer which will accurately control the frequency of very low frequency oscillatory circuits.

A still further object of my invention is to provide a transducer whose resonant frequencies may be varied by adjustment.

A still further object of my invention is to provide a transducer which may be employed as `the active element in a microphone.

A still further object `of my invention is to provide a transducer which may be employed as the -active element of a loudspeaker or tweeter.

A still further object of my invention is to provide a transducer which may be employed as the active element in a phonograph or transcription pickup.

A still further object of my invention is to provide a transducer which may be employed as the active element in a phonograph or transcription record cutter.

A still further object of my invention is to provide a transducer which may be employed as an electromechanical lter.

A still further object of my invention is to provide a transducer which may be employed as a relay.

A still further object of my invention is to provide a transducer which may be employed as the active element in a vibrator or chopper.

A still further object of my invention is to provide a transducer which may be employed as the active element in a hydrophone.

A still further object of my invention is to provide a transducer which may be employed as the active element in a subaqueous sound projector.

A still further object of my invention is to provide a transducer which may be employed as the active element in an accelerometer.

A still further object of my invention is to provide a transducer which may be employed as the active element in a displacement gauge.

A still further object of my invention is to provide a transducer which may be employed as the active element in a velocity gauge.

A still further object of my invention is to provide a transducer which may be employed as the active element in a sonic cleaning or chemical emulsification system.

Other objects and advantages of my invention will be apparent during the course of the following description.

In the accompanying drawings, forming a part of this application, and in which like numerals are employed to designate like lparts throughout the same,

FIGURE 1 illustrates a method presently used to obtain motion of a ilat transducer in exural mode,

3,168,623 Patented Feb. `2, 1965 FIGURE 2 illustrates a method presently used to obtain motion of a cylindrical piezoelectric or polarized electrostrictive transducer in flexural mode,

FIGURE 3 illustrates a method presently used to obtain motion of a magnetostrictive cylindrical transducer in exural mode,

FIGURE 4 is a section of the transducer of FIGURE 3 along ythe line 4-4,

FIGURES 5A, 5B, 5C, 5D and 5E illustrate driving field intensities in transducers of varying characteristics,

FIGURE 6 serves to illustrate and explain the theory underlying my invention, A

FIGURES 6A and 6B illustrate homogeneous and nonhomogeneous eld distributions in magnetostriotive transducers,

FIGURES 7A and 7B illustrate cross-sections of transducers, made of homogeneous material, and which may be driven electrically in flexural mode by applying the driving volta ge on only two electrodes, t

FIGURES 7C and 7D illustrate cross-sections of transducers, made of homogeneous material, and which may not be driven electrically in flexural mode by applying the driving voltage on `only two electrodes,

FIGURE 8 illustrates an embodiment of my invention,

FIGURE 9 is a section of the transducer of FIGURE 8 along the line 9 9,

FIGURES 10 and l1 illustrate further embodiments of my invention utilizing piezoelectric or polarized electrostrictive material,

FIGURE 12 illustrates my invention as embodied in a cylindrical magnetostrictive transducer,

FIGURE 13 is a section of the transducer of FIGURE 12 along the line I3-13,

FIGURES 14 and 15 are employed to explain the theory and operational characteristics of transducers produced according -to my invention,

FIGURE 16 is an illustnati'on of the equivalent electrical circuit of transducers produced in accordance with my invention,

FIGURE 17 illustrates a method of lowering .and continuously adjusting the resonant frequency of transducers produced in accordance with my invention,

FIGURE 18 illustrates a further method of lowering the resonant frequency of transducers produced i'n accordance with my invention,`

FIGURES 19 and 20 illustrate transducers produced in accordance with my invention,

FIGURE 21 illustrates an embodiment of my invention which may be employed as the resonant element of a low frequency oscillator (this transducer may be of the tunable type),

FIGURE 22 illustrates a transducer produced in accordance with my invention, employed as a band-pass iilter (this transducer may be of the tunable type),

FIGURE 23 illustrates the characteristic frequency response `of the filter of FIGURE 22,

FIGURE 24 illustrates the employment of two transducers, produced in accordance with my invention, as a yband-pass lter with adjustable band width,

FIGURE 25 illustrates the characteristic frequency response of the lter of FIGURE 24,

FIGURE 26 illustrates a transducer, produced in accordance with my invention, employed as a low-frequency, high-amplitude vibrator (this transducer may be of the tunable type),

FIGURES 27 and 28 illustrate schematic diagrams of oscillator circuits in which transducers, produced in accordance with my invention, serve as the frequency control elements (these transducers may be of the tunable 0 type),

FIGURE 29 illustrates a twister type transducer, produced in accordance with my invention,

FIGURE 30 illustrates a transducer` produced in accordance with my invention, employed as the active element in a phonograph pickup,

FIGURES 31A and 31B are two possible embodiments of a section of the phonograph pickup of FIGURE 30 along the line 31-31 of FIGURE 30,

FIGURE 32 is a further embodiment of my invention employed as a phonograph pickup,

FIGURE 33 is a plan view and FIGURE 34 is an end view of a torsional type transducer which is produced in accordance with my invention and which requires only two driving electrodes.

FIGURE 35 is a plan view and FIGURE 36 is an end view of a further embodiment of torsional type transducer which is produced in accordance with my invention which is produced in accordance with my invention and which requires only two driving electrodes,

FIGURE 37 illustrates a transducer with a continuously adjustable resonant frequency which is produced in accordance with my invention and which may be employed as the tunable element in oscillators, lters, vibrators and like devices,

FIGURES 38 and 39 illustrate embodiments of my invention employed as the active elements in accelerometers,

FIGURE 40 illustrates a transducer, produced in accordance with my invention, employed as the active element in a h vdrophone,

FIGURE 41 illustrates a microphone utilizing a transducer, produced in accordance with my invention, as the active element therein,

FIGURE 42 illustrates a transducer, produced in accordance with my invention, which may be employed as the active element in a microphone, hydrophone, projector, loudspeaker or tweeter,

FIGURES 43 and 44 illustrate the employment of transducers, produced in accordance with my invention, as relays or vibrators,

FIGURES 45, 46 and 47 illustrate methods of lowering the resonant frequency of transducers produced in accordance with my invention,

FIGURE 48 illustrates a further embodiment of my invention wherein the displacement of the free end of the transducer is large for a given driving voltage input,

FIGURE 49 illustrates a transducer, produced in accordance with my invention, employed as a chopper,

FIGURE 50 is a plot of the voltage characteristic curve of the chopper of FIGURE 49,

FIGURE l illustrates a transducer, produced in accordance with my invention, employed as a chopper, and

FIGURE 52 is a plot of the voltage characteristic curve of the chopper of FIGURE 5l.

In the drawings, wherein for the purpose of illustration. are shown preferred embodiments of my invention, the numeral 100 designates a piezoelectric or polarized electrostrictive transducer with two portions 10011 and 1001), both bonded to a center electrode 102. The outer surface of 100a is bonded to electrode 101 and the outer surface of 100!) is bonded to electrode 103, such that there is good electrical contact between the portions 10011 and 100i) and the electrodes 101, 102 and 103. The combination is suitably attixed to a substantially rigid element 104. However. clamping of the transducer is not essential to its operation and there are many other mounting means which may be employed with equal effect` Terminals 105. 106, and 107 are electrically' connected to electrodes 101, 102, and 103, respectively. If transducer 100 is polarized in the directions shown by the small arrows within 100i: and 100/2, flexural motion will be produced in 100 when voltage is applied bewtcen terminals 105 and 107. If the polarization of 140011 is in the same direction as that of 10011, tlexure of transducer 100 may be obtained by electrically connecting terminals 105 and 107 and applying the voltage between their juncture and terminal 106. If the polarizations of 100a and 100b are as shown in FIGURE l and a direct current voltage is applied between terminals and 107 as shown in the figure, transducer 100 will flex and remain in the llexed position after the voltage source is removed and until the charge leaks off. The direction of the,initial tlexure is determined by the polarity of the applied voltage and will be in one of the directions indicated by arrow 108. Conversely, under the above described conditions of polarization directions, a voltage will appear at the terminals designated for each condition of polarization and electrical connection when transducer 100 is subjected to an applied flexural mechanical force or moment in the direction of arrow 108.

Other methods for making the surfaces of the transducers of my invention electrically conductive, such as reducing the surface by chemical or other means, may be employed in lieu of bonding electrodes to the transducers surfaces. Any method, which produces good electrical contact with the surface of the transducer, may be employed with transducers produced in accordance with my invention. References to electrodes throughout this specicaton also include conductive surface materials obtained by means other than deposition of metallic conductive material on the surface of the transducer.

The numeral 109 designates a cylindrical piezoelectric or polarized electrostrictive transducer with two pairs of electrodes, electrodes 110 and 111 being one such pair and electrodes 112 and 113 being the other pair. The numeral 114 designates the electrical connections from electrodes 110 and 112 to terminal 115 and the numeral 116 designates the electrical connections from electrodes 111 and 113 to terminal 117. Transducer 109 is suitably aixed to substantially rigid element 11S. However, clamping of the transducer is not essential to its operation and there are many other mounting means which may be employed with equal effect. The polarization of transducer 109 is in the direction shown by the small arrows on transducer 109. Reversal of this polarization by 180 degrees will produce the same exural results as herein described. If the alternating voltage is applied between terminals and 117, transducer 109 will ex in the directions indicated by arrows 119 and 120. If a direct-current voltage is applied between terminals 115 and 117 as shown in the figure, transducer 109 will flex and remain in the exed position after the voltage source is removed and until the charge leaks off. Conversely, under the above-described conditions of polarization di rection, voltage will appear across terminals 115 and 117 when transducer 109 is subjected to an applied exural mechanical force or moment in the directions of arrows 119 and 120.

Transducers 100 and 109 will tiert at a rate determined by the frequency or time dependence of the alternating voltage applied to their respective electrical terminals. Conversely, the voltage appearing across said electrical terminals will be of the same frequency or time depend ence as that of mechanical force or moment applied to the transducer.

Cylindrical magnetostrictive transducer 121 is suitably atiixed to substantially rigid element 124. However, clamping of the transducer is not essential to its operation and there are many other mounting means which may be employed with equal effect. Transducer 121 will ex in the directions indicated by arrow when an alternating current tlows thorugh opposed windings 122 and 123, provided there is a direct current ilow through bias windings 12211 and 123a. The time dependence of this tlexure is determined by the frequency or time dependence of the flowing alternating current. Conversely, alternating currents will flow through opposed windings 122 and 123 if there is a direct current flow in bias windings 122a and 123a when transducer 121 is subjected to mechanical force or moment in the directions indicated by arrow 125. The frequency or time dependence of such current tiow will be determined by that of the mechanical force or moment.

The flexural modes of a transducer of a given material, shape and dimensions are of lower resonant frequencies than those of the other modes and the methods illustrated in FIGURES 1, 2, 3, and 4 are those which are presently employed when it is necessary to make use of the lower resonant frequencies of the transducer.

Transducers produced according to my invention will vibrate in exural mode, torsional mode or twisting mode. The torsional and twisting effects are obtained by utilizing the exural motion of each elementary portion of specially shaped transducers and provide very low resonant frequencies of vibration for transducers of very small dimensions. In particular, transducers of the shapes and physical characteristics illustrated inl FIGURES 33 through 36 and 48 will vibrate at the lowest possible resonant frequencies for a given outside dimension. For a given material and set of dimensions, transducers, produced according to my invention will vibrate at lower resonant frequencies than may be obtained by any other method.

The necessary and sutiicient conditions for the production, by the use of electric or magnetic fields, of flexural vibration of a transducer composed of a completely homogeneous material and having only twoy driving electrodes or a single driving Winding are: first, the applied (driving) dynamic field (electric or magnetic) in the cross-section of the transducer shall not be homogeneous (shall be non-homogeneous) over the entire cross-section; and second, the absolute instantaneous values of the field (electric or magnetic) at symmetrical points about the neutral axis shall not be equal. My invention directs itself to those transducers which meet these conditions.

The neutral line in a transducer is that line which does not change its length during ilexure of the transducer. The neutral surface is the surface which contains the neutral line and is perpendicular to the direction of ilexure. The intersection of the neutral surface with the crosssection of the transducer is the neutral axis.

Although there are many ways to obtain non-homogeneous dynamic fields in piezoelectric, polarized electrostrictive or magnetostrictive transducers, such as the use of more than two electrodes or a single driving winding, non-homogeneous material, and like methods, my invention teaches the obtaining of non-homogeneous dynamic fields by the correct and proper shaping of the transducer. Only two driving electrodes are required for a piezoelectric or polarized electrostrictive transducer and only a single driving winding is required for a magnetostrictive transducer.

Transducers shaped and produced in accordance with my invention will contain non-homogeneous dynamic driving fields even though the material comprising the transducer is homogeneous in all respects.

For the electric iield, consider the law of the conservation of flux. We have div E equal to kp; where p is the density of charge, k is a constant, and E is the electric field. Further, the field is always perpendicular to an equipotential surface. The electrodes of the transducer are equipotential surfaces as long as they are much smaller in size than the wavelength of the applied alternating Voltage. For the usual sizes of transducers with electrodes of less than 100 cm. between any two points thereon and applied frequencies of less than megacycles, the electrodes are equipotential surfaces. When the electrodes are curved, the equipotential surfaces are curved and the applied (dynamic) driving field is no longer homogeneous.

For the magnetic tield, consider the Ampere-Stokes theorem in conjunction with the expression (valid at low frequencies), curl B is equal to kj; where B is the magnetic induction, k is a constant :and i is the current density. Transducer 437, shown in FIGURE 6B, contains enclosed area A. Disregarding eddy currents, which may be reduced considerably by conventional methods, there is no current ow anywhere in transducer 437 and consequently, there is no current ow in area A. The dynamic driving field is produced by helical windings 438 which are evenly distributed along transducer 437. Curl B is equal to zero everywhere in 437 and within enclosed area A.

f Bds=ff cundo Portions 2-3 and 4-1 of 437 are such that Bds is equal to zero. (B perpendicular to ds.) We obtain (l-2)Bl- (34)B2 is equal to zero. From which, Bl/Bz is equal to 3-4/1-2; and if 3-4 is not equal to 1-2, then B1 is not equal to B2 and the field is non-homogeneous. FIGURE 6B is a longitudinal section of transducer 347.

FIGURE 6A shows a longitudinal section of magnetostrictive transducer 435 which is of homogeneous material. Longitudinal axis 435.11 is a straight line and the cross-section of transducer 435 is symmetrical to some line perpendicular to 435a. In transducer 435, 3-4 is equal to 1-2 and, therefore, Bl is equal to B2 and the field in 435 is homogeneous.

The driving dynamic instantaneous field 126 of FIG- URE 5A is homogeneous and therefore, there will be no ilexure, electro-mechanically produced, in a transducer of this cross-section and of homogeneous piezoelectric, polarized electrostrictive or magnetostrictive properties if only two driving electrodes or a single driving winding are used. The driving dynamic fields (instantaneous) 127 and 128 of FIGURES 5B land 5C are not homogeneous and if they are not symmetrical about the neutral axis of transducers of this cross-section and of homogeneous material and of homogeneous piezoelectric, polarized electrostrictive or magnetostrictive properties, the transducers can be excited in flexural vibration, by electromechanical means.

In FIGURE SD, magnetostrictive transducer is suitably aixed to substantially rigid body 138 and has windings 136 and 137 wound around the outside; 136 being the driving winding and 137 being the bias winding. The neutral axis is designated as 139 and the driving dynamic instantaneous tield is as indicated by the arrows in the figure. Under these conditions transducer 135 will not vibrate in ilexural mode if it is of homogeneous material.

In FIGURE 5E, magnetostrictive transducer 140 is of homogeneous material and is suitably affixed to substantially rigid body 143. The driving winding is designated by 141, the bias winding is designated by 142 and the neutral axis is designated by 144. The driving dynamic instantaneous field is as indicated by the arrows in the figure and transducer will vibrate in flexural mode.

FIGURE 6 serves to illustrate the positions of the neutral line, neutral surface, and neutral axis with respect to the direction of exure. The numeral 130 designates the neutral line, the numeral 131 designates the neutral surface, the numeral 132 designates the neutral axis, the numeral 133 designates a substantially rigid body to which transducer 129 is suitably affixed and the numeral 134 designates the direction of ilexure. Transducers produced according to my invention need not be rigidly attached at any particular point but may be suitably held at many different points without adversely affecting the operation of the transducers in electro-mechanical flexural vibration.

In FIGURES 7A and 7B, fields 145 and 146 are not homogeneous and are not symmetrical about their respective neutral axes 145a and 146o and, therefore, exure can be produced with only two electrodes in transducers of homogeneous material and of such cross-sections and applied field distributions. Fieid 147, shown in FIGURE 7C, is homogeneous and symmetrical about neutral axis 147a and, therefore, tiexure cannot be produced by electro-mechanical means in a homogeneous transducer of this '7 cross-section and driving field distribution if only two electrodes are employed. Field 148, shown in FIGURE 7D, is not homogeneous but it is symmetrical about neu tral axis 1480 and, therefore, tiexure cannot be produced by electro-mechanical means in a homogeneous transducer f this cross-section and driving field distribution it only two electrodes are employed.

My invention is concerned with obtaining non-homogeneous distribution of the applied driving dynamic field and its non-symmetrical distribution about the neutral axis so that transducers of homogeneous material may be caused to vibrate in flexural mode with only two driving electrodes or a single excitation winding. No matter how the driving dynamic field distributions (electric or magnetic) are obtained, tiexural vibration will be obtained by electro-mechanical means as long as the necessary and sufficient conditions of non-homogeneity of eld distribution and non-symmetry about the neutral axis are fulfilled. The driving dynamic field distributions illustrated throughout the drawings are instantaneous field distributions which vary in accordance with the alternating voltage applied. However, if the instantaneous field distribution satisfies the requirements of iiexure at any one instant, it will satisfy the conditions at all instants, the time and space dependence of such fields being independent functions which are multiplied in the steady state expression of these fields.

FIGURES 8, 9, 10 and ll illustrate some embodiments of my invention as applied to piezoelectric or polan ized electrostrictive transducers of homogeneous material. Transducer 149 will ex in the directions indicated by arrow 154 if an alternating voltage is applied across terminals 152 and 153 which are electrically connected to electrodes 150 and 151 respectively. The triangular cross-section is illustrative of the method of obtaining a nonhomogeneous distribution of the applied dynamic driving eld which is non-symmetrical about the neutral axis of the transducer. Other cross-sectional shapes such as regular and irregular polygons and non-symmetrical curved shapes of various curvatures may be employed equally effectively.

1 choose to define transducers of the type depicted by transducer 155 of FIGURE 10 as split cylinders, a split cylinder being a cylinder with a small portion parallel to the longitudinal axis removed. Transducer 155 will vibrate in flexural mode in the directions indicated by arrows 160 if an alternating voltage is applied across terminals S and 159 which are electrically connected to electrodes 156 and 157 respectively.

Transducer 161, which is a segment of a cylinder, will vibrate in Tiexural mode in the directions indicated by arrows 166 if an alternating voltage is applied across terminals 164 and 165 which are electrically connected to electrodes 162 and 163 respectively. Transducer 161 may also be considered as a split cylinder and it is possible to drive it in a iiexural vibration wherein the displacements are distributed in the manner depicted in FIGURE l0. Portions of other types of curved surfaces will vibrate in similar fashion to that described for transducer 161.

Conversely, since all transducers are electro-mechanical devices, an alternating voltage will be developed across the terminals which are electrically connected to the electrodes when n mechanical stress is applied in the direction of tiexural vibration. The time dependence of the alternating voltage. thus produced. is determined by the time dependence of the mechanical flexure, just as the time dependence of mechanical iiexure is determined by the time dependence of the applied voltage.

Transducer 167 is a split cylinder of the magnetoslrictive type with driving winding 168 and bias winding 169. Numeral 17() designates a block of high permeability magnetic material which is employed to close the magnetic circuit, that is, to decrease the reluctance of the circuit. A direct-current biasing current is caused to flow through 169 and transducer 167 will vibrate in flexural mode in the directions indicated by arrows 171 when an alternating current is made to flow through winding 168. Conversely, so long as the bias current is maintained, an alternating current will fiow in winding 168 when a vibrating mechanical force is applied to transducer 167 in the direction of arrows 171.

As I have stated previously, my invention applies to transducers of the piezoelectric, polarized electrostrictive and magnetostriclive types and the illustration of a particular shape in one of the types but not in the others does not mean that the other types may not be produced in that particular shape. It should be clearly understood that transducers of all these types may be made in all the shapes discussed. While it is not necessary to the operation of my invention, it should be noted that in all the cases discussed the material comprising the transducer is completely uniform and homogeneous. rfhe necessary and sufficient conditions for vibration in tiexural mode with only two driving electrodes or a single driving winding apply when the material of the transducer is uniform and homogeneous. If the material is non-homogeneous, the conditions may be sufficient but not necessary.

In order to further explain the theory underlying my invention7 it is advisable to consider transducer 172 of FIGURE 15 which has a length equal to 2l and a thickness to t. Assume that the curve along 2l is smooth, that the cross-section is the same throughout the transducer and that ZI is greater than t. Then the resonant frequencies in fi-exural mode f are given by the expression'.

z2 /EI 0 27121 ps' where zznumber depending on the harmonic considered and the type of support of the transducer (2:2.635 for the fundamental flexural mode with the transducer unclaniped) EzYoungs modulus ofthe material being used lzmoment of inertia of the cross-section with respect to the neutral axis of the cross-section pzspecific gravity l=half the length of the flexing transducer szcross-section of the exing transducer.

For a given mode (either fundamental or harmonic) and a given material. E. and p are constants and 1:1615, where k is the radius of gyration, we obtain the expression:

F=eoustant 7,2: constant Fl=constant In the case shown in FIGURE 14, in which transducer 173 is a split cylinder or in any other case in which the cross-section 174 is a rectangle, we obtain the expression:

l k I'; V12

where l is the cylinder wall thickness; from which Z2 F-t constant For barium titanate. the constant is found to be equal to approximately l.lt) 105 cm./sec.

Therefore. for a given resonant frequency, an innite number of values of l and t exist which will satisfy the equation. For example, given a barium titannte transducer of the split cylinder shape for which a liexural resonant frequency of cycles per second is desired. substitution in the equation gives:

t=0.l cm. t=0.25 cm. and so forth.

FIGURE 16 shows the electrical equivalent circuit of a freely vibrating piezoelectric or polarized electrostrictive transducer, where numeral 175 represents the static capacitance of the transducer, 175e represents the electromechanical coupling, 176 represents the motional inductance, 177 the motional capacitance, and 17S the motional resistance. T o match the transducers equivalent electrical impedance to the electrical generator output impedance, it is useful to add an inductance in series or parallel with the transducer. This matching is especially useful in applications in which the transducer must be driven at relatively low voltage for large amplitudes of vibration.

Split cylinders have a very high figure of merit, Q, of the order of 1,000 to 3,000 at frequencies from 20 to 1,000 cycles per second. This is much higher than circuits, employing purely electrical components, yield at these low frequencies. The value of Q may be varied 4by clamping the transducer, or by adding mechanical loads, or by immersing all or part of the transducer in a liquid, or by increasing the friction or by similar means. The lowest value of damping is obtained when the transducer is clamped at the nodal surfaces and the electrode leads are connected on the nodal surfaces. The position of the nodal surfaces varies with the shape of the transducer and may be calculated from the particular solution of the differential equation of linear flexure corresponding to the particular mode and type of boundary conditions under study. The Q of a given transducer may be adjusted to almost any value between 0.5 and 3,000 (approximately). The resonant frequency for a given transducer of a particular material and dimensions will vary by approximately at these extreme values of Q.

FIGURES 17 and 18 illustrate two methods of lowering the flexural mode resonant frequency of a given transducer. Transducer 179 has securely aixed thereto threaded elements 180 on which are threaded masses 181. The further away masses 181 4are from transducer 179, the more the resonant frequency of the tranducers flexural mode is displaced from the original resonant frequency. The resultant resonant frequency is lower or higher than the original resonant frequency depending on the direction of threaded elements 180 with respect to the motion of transducer 179 in exurai vibration.

Transducer 182 has axed thereto masses 183. These masses 183 are not adjustable and the flexural resonant frequency is likewise xed. However, the frequency is lower than it would be if the same transducer were unloaded.

Transducers 184 and 186 are produced according to my invention and will vibrate in flexural mode when driven by only two electrodes. In the case of 184, the electrodes are designated 188 and 189 and in the case of 186, the electrodes are designated 190 and 191. The nodal surfaces of transducer 184 are designated by numeral 185 and those for transducer 186 are designated by numeral 187.

Transducer 192 may be employed as the frequency control element of a low frequency oscillator. Numeral 193 designates the driving electrode, numeral 194 designates the feedback electrode and numeral 195 designates the electrode which is maintained at ground potential. The various methods of adjusting the resonant frequency of the flexural vibrations of the transducers which are described in this specication may be employed to adjust the frequency of transducers of the type of transducer 192.

Transducer 196 is shown connected to serve as a bandpass lter wherein the input is applied between electrodes 197 and 199 and the output is obtained across electrodes 198 and 199. Minimum damping and highest Q is ob- 10 tained if the clamping of transducer 196 and the electrical leads for the input and output connections are accomplished at the nodal surfaces 200. Curve 201 is a typical output/input against frequency plot of the response of transducer 196.

Transducers 202 and 203 are shown connected as a band-pass lter with adjustable resonant frequencies. Masses 204 and 20S may be moved closer to or further away from transducers 202 and 203 in order to vary the resonant frequency of each transducer. Curve 206 is a typical output/input against frequency plot of the response of the combination of transducers connected as shown in FIGURE 24.

Transducer 207 is shown connected as a low-frequency high-output vibrator. Coil 208 serves to balance out the static capacitance of transducer 207 and to match the equivalent electrical impedance of transducer 207 to the output impedance of generator 209 at the resonant frequency. The frequency of generator 209 is adjusted to the resonant frequency of the fiexural mode at which we desire to have transducer 207 vibrate.

Transducers 210 and 211 are shown as the control elements of typical electronic oscillator circuits. Because of the low resonant frequency of these transducers in flexural mode, it is possible to obtain accurate frequency control at very low frequencies with simple electronic circuits, such as are shown in FIGURES 27 and 28.

Transducer 212 is a twister type transducer in which the flexural vibrations of all elementary portions and the shape of the transducer produce the twisting motion. Transducers of the type of transducer 212 are useful as microphone elements. The electrodes are applied to faces 214 and 215 and when an alternating voltage is applied between them, transducer 212 will vibrate as indicated by arrows 213.

Transducer 216 is a transducer, produced in accordance with my invention, employed as the active element in a phonograph pickup or cutter. One end of transducer 216 is clamped to substantially rigid body 217 so that when pickup needle 218 vibrates, avoltage appears across electrodes 219 and 220. Conversely, if a voltage is applied across electrodes 219 and 2213, needle 218 will vibrate in accordance with the time dependence of the applied voltage. FIGURE 31A illustrates a single type pickup or cutter needle 218 and FIGURE 31B illustrates a dual type pickup needle which may be employed for use with phonograph records and transcriptions which require elements of different tip radius. When the dual pickup element, as illustrated in FIGURE 31B, is employed, transducer 216 is pivoted about point 221 so that either pickup needle 218-1 or 218-2 is in Contact with the record groove or transcription groove which is to be reproduced by the system.

Transducer 222 is the active element in a-dual needle type of phonograph or transcription pickup to which needles 223 and 224 have ben suitably alhxed. The combination is pivoted within clamp 229 so that the required needle is in contact with the record or transcription to be reproduced. Electrical output connections are made to terminals 227 and 228 which are electrically connected to electrodes 225 and 226 respectively.

Transducer 230 is a torsional type-transducer in the shape of a cylinder with electrodes 231 and 232 on the surfaces of said transducer and a helix 233 cut along the surface of 230. When an alternating voltage is applied across electrodes 231 and 232, transducer 230 will twist in the directions indicated by the arrows 234.

Transducer 235 is similar to transducer 230 except that helix 238, which is cut in transducer 235, does not extend along the whole length of transducer 235. If an alternating voltage is applied between electrodes 236 and 237, transducer 235 will twist in the directions indicated by arrows 239.

Transducer 240 is a tunable very low frequency resonant element which may be employed in all applications which require a tunable element. Transducer 240 is clamped between rollers 241 and 242 which are suitably fixed in holder 244. Rollers 241 and 22 are coated with a material 2410, the stiffness of which is chosen to permit obtaining the mechanical Q which is desired. This coating material may be made electrically conductive or it may be insulating, according to the requirements. lf they are conductive, they may be used for the electrical connections to electrodes 245 and 246 with which they are in contact. The clamping action of rollers 241 and 242 fixes the nodal lines at their line of contact with transducer 240 and this in turn, fixes the resonant f requency, as determined by the distance between the lines of clamping and the free ends of transducer 240. By adjusting knob 243, the position of clamping is shifted on transducer 240 which moves between rollers 241 and 242, and thus, the resonant frequency is adjusted. External electrical connections are made (when 241e is of insulating7 material) to terminals 247 and 248 which are electrically connected to electrodes 245 and 246, respeo tively.

Transducer 24:9 is the active element in an accelerometer or strain gauge. Mass 251 is applied to one end of 249 and the other end is clamped to substantially rigid body 250. In some applications, mass 251 is not required, in which case the mass of the transducer 249 itself serves as the inertial element. External electrical connections are made to terminals 254 and 255 which are electrically connected to electrodes 252 and 253, respectiveiy.

Transducer 256 is the active element in an acceierometer, strain gauge or pressure gauge. Mass 257 is afxed to one end of transducer 256 while the other end is suitably clamped to substantially rigid body 258. External connections are made to terminals 261 and 262 which are electrically connected to electrodes 259 and 260, respectively.

Transducer 263 is a subaqueous hydrophone or projector. Electrodes 26-2` and 265 are respectively connected to terminals 266 and 267 to which the outside electrical connections are made. Membranes 268 serve to contain liquid 269 which is employed to damp the vibration of 263. In some applications. only a single membrane is required, in which case, it serves to prevent the outer liquid from entering the transducer. The numeral 270 designates an air space which is sealed from the outer liquid by membranes 268 and at both ends of 263. Clamps 271 are applied to transducer 263 along the nodal surfaces and are suitably atixed to substantially rigid body 272.

Transducer 273 is the active element of a microphone. One end of 273 is attached to pin 274 which in turn is attached to diaphragm 275. The other end of 273 is clamped at point 276 to housing 277. When sound waves strike diaphragm 275, pin 274 causes transducer 273 to vibrate in tlexural mode and an electrical voltage appears across terminals 280 and 281 which are electrically connected to electrodes 278 and 279, respectively. Conversely, the same embodiment may be employed as a loudspeaker or tweeter. In which case, the application of alternating voltage at audio frequencies across terminals 280 and 281 will cause diaphragm 275 to vibrate at audio frequencies.

Transducer 282 in the embodiment illustrated may be employed as a microphone, hydropltone, projector, loudspeaker or tweeter. The numeral 283 designates an acoustic isolator such as cork. corprcne or like material which serves to prevent transmission or reception of sound in the inner portion of the transducer 282. Electrical connections are made to terminals 286 and 287 which are electrically connected to electrodes 284 and 285, respectively.

Transducer 288 is illustrated as a relay wherein terminals 293 and 294 are rigidly attached to one end or" 283 and malte contact with terminals 292 and 295 during flexure. When contact is closed between terminals 292 and 293, it is open between 294 and 295 and vice versa. Transducer 288 is caused to ex by the application of a voltage across terminals 296 and 297 which are electrically connected to electrodes 290 and 291 respectively. The end of transducer 288 carrying terminals 293 and 294 will have the maximum displacement amplitude because the other end is suitably aixed to substantially rigid body 289.

Transducer 298 is a relay which closes two circuits when it is flexed in one direction and opens the circuits when it is iiexed in the opposite direction. Transducer 298 is caused to ex when a voltage is applied across terminals 301 and 302 which are electrically connected to electrodes 299 and 300, respectively. When transducer 298 tlexes outward, contacts 303 and 305 are electrically connected and contacts 304 and 306 are electrically connected. When the transducer 298 lexes inward, the contacts are broken and the circuits are opened. More than two sets of contacts may be opened and closed in this manner since a plurality of such contacts may be axed to the transducer along the flexing cross-section. The relays depicted in FIGURES 43 and 44 may be operated with either a D.C. or an A.C. voltage. The maximum amplitude of displacement is attained when the AC. frequency is equal to the transducers exural resonant frequency.

Transducers 307, 308, and 309 are illustrative of various methods of increasing the mass-compliance of transducers of given dimensions in order to lower the resonant frequencies. These transducers may be employed as the active elements in any of the applications for which the transducers contemplated by my invention may be used.

Transducer 310 is clamped to substantially rigid body 315 and will flex in the directions indicated by arrow 316 when a voltage is applied across the 4terminals 313 and 314 which are electrically connected to electrodes 311 and 312, respectively. The motion will `be of particularly large amplitude for transducers of small dimensions.

Transducer 317 is clamped at its nodal surfaces 325 and is employed as a chopper. The time dependence of its output waveform across terminals 326 is illustrated by curve 327. Curve 327 is typical of the output when a direct-current voltage is applied across terminals 324. With no voltage applied to terminals 324, terminal 319, which is bonded to electrode 318, is in Contact with terminal 320, and terminal 322, which is bonded to electrode 321, is in contact with terminal 323. When a direct-current voltage of the proper polarity is applied across terminals 324, the transducer 317 tiexes inward and breaks the contacts between terminals 319 and 320 and between terminals 322 and 323. After the charge is removed because of current ow in the circuit associated with terminals 326, the transducer 317 returns to its original position and the cycle starts again.

Transducer 328 is a chopper with electrodes 329 and 330. Terminals 331 and 332 are bonded to electrodes 329 and 330, respectively. In the unexed position, tcr minals 331 and 332 are in contact with spring contacts 333 and 334 respectively. When a direct-current voltage of the proper polarity is applied to terminals 335, spring contacts 333 and 334 remain in contact with terminals 331 and 332 for a longer period than if the contacts 333 and 334 were not on springs and the consequent output waveform across terminals 336 is of the shape shown by curve 337. After the contact between 333 and 331 and between 332 and 334 is broken, the charge is removed from transducer 32S because of the current How in the electrical circuit associated with terminals 336 and transducer 328 returns to its original position and the cycle starts again.

While I have described my invention by means of specific examples and in specific embodiments. I do not wish to be limited theretolor obvious modifications will 13 occur to those skilled in the art without departing from the spirit and scope of the invention.

Having thus described my invention, I claim:

1. A transducer comprising a homogeneous electromechanically sensitive body, said body being a hollow cylinder carrying a slit therein parallel to the axis of said cylinder, the surfaces forming the sides of said slit being substantially uniform in cross-section, at least one electric coil, and retaining means in the surfaces of said cylinder for receiving said coil.

2. A transducer comprising an electro-mechanically sensitive body, said body being a hollow cylinder and carrying a helical slit therein, said slit traversing the surface of said cylinder, each end of said slit being spaced from the end of said cylinder adjacent thereto, and means for making electrical connection to said transducer.

3. A transducer as described in claim 2 wherein the material comprising said electro-mechanically sensitive body is homogeneous.

4. A transducer as described in claim 2 wherein the material comprising said electro-mechanically sensitive body is a homogeneous ceramic.

5. A transducer comprising an electro-mechanically sensitive body, said body being a hollow cylinder and carrying a helical slit therein, said slit traversing the surface of said cylinder, one end of said slit being spaced from the end of said cylinder adjacent thereto, and means for making electrical connection to said transducer.

6. A transducer as described in claim 5 wherein the material comprising said electro-mechanically sensitive body is homogeneous.

7. A transducer as described in claim 5 wherein the material comprising said electro-mechanically sensitive body is a homogeneous ceramic.

8. A transducer comprising an electro-mechanically sensitive body, said body being a hollow cylinder and carrying a helical slit therein, said slit traversing the surfacing of said cylinder, each end of said slit coinciding with one end of said cylinder, and means for making electrical connection to said transducer.

9. A transducer as described in claim 8 wherein the material comprising said electro-mechanically sensitive body is homogeneous.

10. A transducer as described in claim 8 wherein the material comprising said electro-mechanically sensitive body is a homogeneous ceramic.

1l. A transducer comprising an electro-mechanically sensitive body, said body being a hollow cylinder carrying a slit therein parallel to the axis of said cylinder. the surfaces forming the sides of said slit being substantially uniform in cross-section, a rst resilient roller in contact with the inner surface of said body, a second resilient roller in contact with the outer surface of said body, holding means supporting said rollers, controlling and adjusting means for adjusting the position of said cylindrical body with respect to said rollers, and means for making electrical connections to the surfaces of said cylindrical body.

12. A transducer comprising an electro-mechanically sensitive body, said body being spiral in shape in its long dimension and planar in its short dimensions, a supporting member, one end of said spiral body being atlixed to said supporting member, and means for making electrical connections to the curved surfaces of said body.

13. A transducer as described in claim l2 wherein the material comprising said electro-mechanically sensitive body is homogeneous.

14. A transducer as described in claim 12 wherein the material comprising said electro-mechanically sensitive body is a homogeneous ceramic.

15. An audio transducer comprising a housing, an electro-mechanically sensitive body, said body being a hollow cylinder carrying a slit therein parallel to the axis of said cylinder` the surfaces forming the sides of said slit being substantially uniform in cross-section, a diaphragm, said body being aflixed to said housing along one of the surfaces forming one of the sides of said slit, said body being mechanically connected to said diaphragm adjacent the side of said slit opposite said side afxed to said housing, and means for making electrical connections to said body.

16. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, said body being mechanically displaced in a flexural mode upon the application of an electrical signal to the electrodes.

17. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes,` said body producing electrical signals at said electrodes upon mechanical displacement of said body in a flexural mode.

18. A transducer comprising an electro-mechanically sensitive body, saidbody being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said body with at least one of the side edges thereof free to move, said body being mechanicmly displaced in a exural mode, upon the application of an electrical signal to the electrodes, to ex said at least one of said side edges inwardly and outwardly.

19. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on theouter convex surface, said body being polarized radially between the electrodes, means for mounting said body with at least one of the side edges thereof free to move, said body producing electrical signals at said electrodes upon mechanical displacement of said body in a tlexural mode by moving said at least one of said side edges inwardly and outwardly.

20. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezoelectric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and Substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, said body being mechanically displaceable in a exural mode about nodal points in the body, means for mounting said body adjacent said nodal points 4with the side edges of the body free to move, said body `being mechanically displaced in a flexural mode, upon the application of an electrical signal to the electrodes, to flex said side edges inwardly and outwardly.

2l. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezoelectric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially y straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes. said body being mechanically displaceable in a tlexural mode about nodal points in the body, means for mounting said body adjacent said nodal points with the side edges of the body free to move, said body producing electrical signals at said electrodes upon mechanical displacement of said body in a flexural mode by moving said side edges inwardly and outwardly.

22. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially .between the electrodes, means for mounting said body adjacent one of the side edges thereof with the other side edge thereof free to move, said body being mechanically displaced in a flexural mode, upon the application of an electrical signal to the electrodes, to flex the free side edge inwardly and outwardly.

23. A transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic ywith inner concave and outer convex substantially cylindrical surfaces, substantially straight `parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said body adjacent one of the side edges thereof with the other side edge thereof free to move, said body producing electrical signals at said electrodes upon mechanical displacement of said body in a exural mode by moving the free side edge inwardly and outwardly.

24. A transducer comprising an elongated electro-rnechanically sensitive body, said body being a `partial hollow cylinder of piezo-electric ceramic with the inner concave and outer convex substantially cylindrical surfaces, Substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means yfor mounting said elongated body with at least one of the ends thereof `free to move, said elongated body being mcchanically displaced in a exural mode, upon the application of an electrical signal to the electrodes, to flex said at least one of said ends thereof laterally towards the concave and convex surfaces thereof.

25. A transducer comprising an elongated electro-mechanically sensitive body, said body being a .partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, sub stantially straight Aparallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said elongated body with at least one of the ends thereof free to move, said elongated body producing electrical signals at said electrodes upon mechanical displacement of said body in a flexural mode by moving said least one of said ends thereof laterally towards the concave and convex surfaces thereof.

26. A transducer comprising an elongated electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said elongated body adjacent one end thereof with the other end thereof free to move, said elongated body being mechanically displaced in a flexural mode, upon the application of an electrical signal to the electrodes, to flex the free end thereof laterally towards the concave and convex surfaces thereof.

27. A transducer comprising an elongated electro-rnechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic twith the inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said elongated body adjacent one end thereof with the other end thereof free to move, said elongated body producing electrical signals at said electrodes upon mechanical displacement of said body in a flexural mode by moving the free end thereof laterally towards the concave and convex surfaces thereof.

28. A transducer for a phonograph comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said body, and a stylus affixed to the body for mechanically displacing said body in a flexural mode for producing electrical signals at said electrodes in accordance with the flexing of the body by the stylus.

29. A transducer for a phonograph comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of piezoelectric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said body/ adjacent one of the side edges thereof, and a stylus afiixcd to the body adjacent the other side edge thereof for mechanically displacing said body in a flexural mode for producing electrical signals at said electrodes in accordance with the flexing of the body by the stylus.

30. A transducer for a phonograph comprising an elongated electromechanically sensitive body, said body being a partial hollow cylinder of piezo-electric ceramic with inner concave and outer convex substantially cylindrical surfaces, substantially straight parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized `radially between the electrodes, `means for mounting said elongated body adjacent one end thereof, and a stylus affixed to the elongated body adjacent the other end thereof for mechanically displacing said body in a flexural mode for producing e1ec.

trical signals at said electrodes in accordance with the flexing of the 4body by the stylus.

3l. An audio transducer comprising an electro-mechanically sensitive body, said body being a partial hollow cylinder of lpiezo-electric ceramic `with inner concave and outer convex substantially cylindrical surfaces, substantially straight .parallel side edges and substantially arcuate ends, an electrode on the inner concave surface, an electrode on the outer convex surface, said body being polarized radially between the electrodes, means for mounting said body, means for applying electrical signals to said electrodes for displacing said body in a flexural mode, and a sound producing diaphragm connected to said body and vibrated by the flexing of said body.

References Cited in the file of this patent UNITED STATES PATENTS 1,750,124 Pierce Mar. ll, 1930 2,101,272 Scott Dec, 7, 1937 2,365,738 Williams Dec. 26, 1944 2,459,019 `DHeedene Jan. 11, 1949 2,515,446 Gravely July 18, 1950 2,719,929 Brown Oct. 4, 1955 2,838,695 Thurston June l0, 1958 

16. A TRANSDUCER COMPRISING AN ELECTRO-MECHANICALLY SENSITIVE BODY, SAID BODY BEING A PARTIAL HOLLOW CYLINDER OF PIEZO-ELECTRIC CERAMIC WITH INNER CONCAVE AND OUTER CONVEX SUBSTANTIALLY CYLINDRICAL SURFACES, SUBSTANTIALLY STRAIGHT PARALLEL SIDE EDGES AND SUBSTANTIALLY ARCUATE ENDS, AN ELECTRODE ON THE INNER CONCAVE SURFACE, AN ELECTRODE ON THE OUTER CONVEX SURFACES, SAID BODY BEING POLARIZED RADIALLY BETWEEN THE ELECTRODES, SAID BODY MECHANICALLY DISPLACED IN A FLEXURAL MODE UPON THE APPLICATION OF AN ELECTRICAL SIGNAL TO THE ELECTRODES. 